#243203

order	512
nilpotency class	3
isoclinism	isoclinic class of #233220
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

634 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 202 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,8)(4,6)(5,7)(9,10,12,16,13,17,20,15)(11,14,19,23,18,22,24,21)(25,26)(27,31)(28,33)(29,32)(30,34)(35,40)(36,38)(37,39);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,18)(15,21)(16,23)(17,22)(20,24)(25,27)(26,31)(28,36)(29,37)(30,35)(32,39)(33,38)(34,40);(1,4,5,3)(2,6,7,8)(10,15)(12,20)(14,21)(16,17)(19,24)(22,23)(26,32)(27,35)(28,30)(31,38)(36,37)(39,40);(11,18)(14,22)(19,24)(21,23)(25,28,29,30)(26,33,32,34)(27,36,37,35)(31,38,39,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,c^2=e,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=dh,g^b=gi,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150399);